Functions of Baire class one

Denny H. Leung ; Wee-Kee Tang

Fundamenta Mathematicae, Tome 177 (2003), p. 225-247 / Harvested from The Polish Digital Mathematics Library

Résumé

Let K be a compact metric space. A real-valued function on K is said to be of Baire class one (Baire-1) if it is the pointwise limit of a sequence of continuous functions. We study two well known ordinal indices of Baire-1 functions, the oscillation index β and the convergence index γ. It is shown that these two indices are fully compatible in the following sense: a Baire-1 function f satisfies $β(f) ≤ ω^{ξ₁} · ω^{ξ₂}$ for some countable ordinals ξ₁ and ξ₂ if and only if there exists a sequence (fₙ) of Baire-1 functions converging to f pointwise such that $supₙβ(fₙ) ≤ ω^{ξ₁}$ and $γ((fₙ)) ≤ ω^{ξ₂}$. We also obtain an extension result for Baire-1 functions analogous to the Tietze Extension Theorem. Finally, it is shown that if $β(f) ≤ ω^{ξ₁}$ and $β(g) ≤ ω^{ξ₂}$, then $β(fg) ≤ ω^{ξ}$, where ξ = maxξ₁+ξ₂,ξ₂+ξ₁. These results do not assume the boundedness of the functions involved.

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Publié le : 2003-01-01

Zbl 1056.26002

EUDML-ID : urn:eudml:doc:283074